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Numerical integration of an erythropoiesis model with
explicit growth factor dynamics
Oscar Angulo, Fabien Crauste, Juan Carlos López-Marcos
To cite this version:
Numerical integration of an erythropoiesis model with
explicit growth factor dynamics
O. Anguloa,∗, F. Crausteb,c, J.C. L´opez-Marcosd
a Departamento de Matem´atica Aplicada. ETSIT. Universidad de Valladolid. Pso. Bel´en 5,
47011 Valladolid. SPAIN. Phone: +34 983 423000(Ext: 5835). FAX: +34 983 423661.
bUniv. Lyon, Universit´e Lyon 1, CNRS UMR 5208, Institut Camille Jordan, 43 blvd du 11
novembre 1918, F-69622 Villeurbanne-Cedex, France
cInria, Villeurbanne, France
dDepartamento de Matem´atica Aplicada. Facultad de Ciencias. Universidad de Valladolid.
Pso. Bel´en 7, 47011 Valladolid. SPAIN.
Abstract
Erythropoiesis, the red blood cell production process, involves interactions be-tween cell populations with different differentiation states, mainly immature progenitor cells and mature erythrocytes, and growth factors such as erythro-poietin and glucocorticoids, known to respectively inhibit cell apoptosis, stimu-late proliferation and differentiation, and stimustimu-late self-renewal. The feedback regulation of this process allows a very fast and efficient recovery in the case of a severe anemia. We consider an age-structured model of red blood cell pro-duction accounting for these feedback regulations and the dynamics of growth factors. We theoretically show the existence of a unique positive steady state for the model and we propose a numerical method to obtain an approximation to its solution. Experiments are reported to show numerically, on one hand, the optimal convergence order of the numerical scheme and, on the other hand, a fine approximation to real experimental data, with a suitable selection of the parameters involved.
Keywords: erythropoiesis model, nonlinear age-structured system, numerical scheme, long-time integration
2000 MSC: 65M25, 35B40, 35L60, 35Q92, 92D25
1. Introduction
Production of red blood cells occurs in the bone marrow, where undifferen-tiated immature hematopoietic stem cells differentiate, throughout a number of divisions, into more and more mature cells until fully differentiated mature red blood cells are produced. The process of production and regulation of red blood cells is called erythropoiesis. The first stage of differentiation of hematopoietic stem cells is the progenitor stage: erythroid progenitors are undifferentiated cells committed to the red blood cell – or erythroid – lineage. These cells share
∗Corresponding author
a lot of properties with stem cells, such as the ability to proliferate and differ-entiate but also to self-renew, that is to produce by division two daughter cells identical to the mother cell. Erythroid progenitors proliferate and differentiate in more mature cells, meanwhile losing their self-renewal ability, until they reach the stage of reticulocytes. These latter are immature red blood cells that will finish their differentiation process within few divisions, ejecting their nucleus, and finally differentiating in erythrocytes that leave the bone marrow and enter the bloodstream as red blood cells. The entire process is regulated by a vari-ety of feedback controls, involving growth factors. Among them, erythropoietin has been identified in 1990 as the main regulator of erythropoiesis: released by the kidneys when the organism detects a lack of red blood cells, erythropoietin binds to the surface of erythroid progenitors and prevents their death, called apoptosis, hence increasing the amount of differentiated cells [24]. Glucocorti-coids have also been shown to influence erythropoiesis in stress conditions by inducing erythroid progenitor self-renewal [15, 22].
We propose a mathematical model of stress erythropoiesis in mice based on age-structured nonlinear partial differential equations describing erythroid pro-genitor and erythrocyte dynamics in the next section. This model is based on previous models, proposed in [8, 21], that were inspired by the initial works by Mackey [26] and co-authors, and Loeffler et al. [25, 33], to describe stress ery-thropoiesis in mice. Several mathematical models of eryery-thropoiesis have been proposed over the last 30 years (see [32] for a review), in order to address the mechanisms of regulation and their role in stress or pathological conditions. Mackey Mackey [26] published a pioneering paper on hematopoietic stem cell dynamics, which has then been used and modified by many authors, including Mackey and co-workers, to describe differentiation and maturation processes involved in hematopoiesis more precisely [4, 5, 18, 19, 31]. Belair et al [16] proposed a model of erythropoiesis considering the influence of growth factors on stem cell differentiation in erythroid progenitors, later improved by Mahaffy et al [27], and recently analyzed in more details [2, 3]. Another erythropoiesis model, inspired by the same article, was introduced by Adimy et al. [6] in 2006, in which Epo is the only growth factor supposed to act during erythropoiesis. An important contribution to mathematical modeling of erythropoiesis is also due to Loeffler and his collaborators [25, 32, 37]. Their models consider feed-back controls from progenitors on the stem cell level and from mature cells on progenitors, and were fitted to various experiments (including irradiations, bleeding, and phenylhydrazine treatments of mice). Most of these works were nevertheless performed before the role of Epo was definitely identified and long before erythroid progenitor self-renewal was hypothesized.
Progenitors
p(t,a) ErythrocytesAgeing
HSC Apoptosis Stem Cells Death Self-renewing Progenitors s(t,a) Epo GC (a,Epo) (Epo) (a,GC) p c A γ(a)
Figure 1: Schematic Model of Erythropoiesis. Plain lines with arrows represent cell pro-cess (differentiation, death, proliferation, cell cycle), whereas dashed lines represent feedback controls.
second order of convergence, and that it allows to compare simulated solutions of the model with real biological data.
2. Model of Stress Erythropoiesis
Let consider a population of erythroid progenitors and a population of ery-throcytes, defined by cell age a and the time of the observation t (see Figure 1). Cell age a is considered to be the time spent from the birth of a cell up to the time of observation, hence cells age with unitary velocity (see (4)). Among progeni-tors, a difference is made between self-renewing progeniprogeni-tors, whose population is denoted by s(t, a), and differentiating progenitors that are not self-renewing, denoted by p(t, a). The duration of the differentiating progenitor compartment is denoted by τp, and the duration of one self-renewing cycle by τc. One may
note that τc < τp. All progenitors are supposed to be localized in the bone
marrow, and all of them can die by apoptosis. The number of erythrocytes, circulating in blood, is denoted by e(t, a), and E(t) denotes the total number of erythrocytes at time t, defined by
E(t) = Z +∞
0
e(t, a)da.
Two growth factors, erythropoietin (Epo) and glucocorticoids (GC), are as-sumed to be the main regulators of erythroid progenitor apoptosis and differen-tiation, and self-renewal respectively [21]. Denote by Epo(t) the concentration of Epo at time t in bloodstream, and by GC(t) the concentration of gluco-corticoids at time t. These concentrations are regulated by the total number of erythrocytes E(t): the more erythrocytes, the less Epo and glucocorticoids [15, 24].
Denote by α(Epo) the erythroid progenitor apoptosis rate, and by σ(a, GC) the erythroid progenitor self-renewal rate. The function α is assumed to be decreasing with respect to Epo (negative feedback, [24]). The function σ is assumed to be increasing with respect to GC (positive feedback), and decreasing with respect to the age a of progenitor cells [15] with σ(τp, GC) = 0. This
contrary to early progenitors that are more sensitive to GC. These assumptions are summarized hereafter, with ∂x denoting the partial derivative with respect
to x:
α0(Epo) ≤ 0, ∂aσ(a, GC) ≤ 0, ∂GCσ(a, GC) ≥ 0, σ(τp, GC) = 0.
Denote by δ(a, Epo) the differentiation rate of progenitors into erythrocytes. This rate depends on the age of progenitors, older ones being more inclined to differentiate [23], and on the concentration of Epo, the more Epo the more differentiation [23]. The function δ is hence assumed to be increasing with respect to a and Epo,
∂aδ(a, Epo) > 0, ∂Epoδ(a, Epo) > 0.
In addition, since all progenitors which did not die in the differentiating compart-ment must differentiate in erythrocytes, we assume lima→τpδ(a, Epo) = +∞
whatever the level of Epo.
The following functions α, σ and δ, will be used in section 5 to perform numerical experiments, α(Epo) = Cα θnα α θnα α + (log10(Epo))nα , (1)
where Cα> 0 is the maximal apoptosis rate, θα > 0 is a threshold value, and
nα> 1 is a sensitivity parameter; σ(a, GC) = Cσ(τp− a) (log10(GC))nσ θnσ σ + (log10(GC))nσ , (2)
where, similarly to the function α, θσ > 0, nσ > 1, and Cσ > 0 is a constant
characterizing the maximum self-renewal rate given, for a fixed value of GC, by Cσ(τp− a); δ(a, Epo) = Cδ1 τp τp− a + Cδ2 (log10(Epo))nδ θnδ δ + (log10(Epo))nδ =: δ1(a) + δ2(Epo), (3)
where Cδ1 is the minimum of the age-dependent differentiation rate δ1(a),
reached when a = 0, Cδ2 is the maximum of the Epo-dependent
differentia-tion rate δ2(Epo), and Cδ1 > 0, Cδ2 > 0, θδ > 0 is a threshold value for the
Epo-dependent part of the differentiation rate, nδ > 1 a sensitivity parameter.
It must be noted that in the absence of GC, the self-renewal rate is sup-posed to vanish, whatever the age of cells, whereas differentiation occurs in the presence or the absence of Epo: Epo is assumed to positively act on erythroid progenitor differentiation, yet differentiation can be induced independently of Epo levels. This explains why the differentiation rate is composed of two addi-tive terms, whereas the self-renewal rate is composed of only one term.
Then, the quantities p, s and e satisfy the following system, for t > 0,
∂tp(t, a) + ∂ap(t, a)=− [α(Epo(t)) + δ(a, Epo(t))] p(t, a)
−σ(a, GC(t))p(t, a), a ∈ (0, τp),
∂ts(t, a) + ∂as(t, a)=−α(Epo(t))s(t, a), a ∈ (0, τc),
∂te(t, a) + ∂ae(t, a)=−γ(t, a)e(t, a), a > 0.
(4)
Boundary conditions associated with (4) describe cell flux between compart-ments, p(t, 0) = HSC + 2s(t, τc), s(t, 0) = Z τp 0 σ(a, GC(t))p(t, a)da, e(t, 0) = A Z τp 0
δ(a, Epo(t))p(t, a)da.
(5)
New erythroid progenitors come both from the division of self-renewing progen-itors in two newborn cells and the differentiation of hematopoietic stem cells: HSC denotes the flux of hematopoietic stem cells differentiating in erythroid progenitors, assumed to be constant over time. Self-renewing progenitors are produced at a rate σ from differentiating progenitors. Erythrocytes are pro-duced from differentiating progenitors, and A denotes a constant amplification coefficient accounting for divisions of mature progenitors. For instance, A = 2n where n is the number of differentiation stages during the reticulocyte stage [21].
Concentrations of growth factors Epo(t) and GC(t) satisfy the following ordinary differential equations [20],
d
dtEpo(t) = f (E(t)) − kEpoEpo(t), d
dtGC(t) = g(E(t)) − kGCGC(t),
(6)
where kEpo > 0 and kGC > 0 are degradation rates, and the functions f and g
describe feedback controls by the total number of erythrocytes on the production of growth factors. Both functions are assumed to be decreasing (see Figure 1),
f0(E) < 0, g0(E) < 0. The model is completed with suitable initial conditions
p(0, a) = p0(a), a ∈ [0, τp], s(0, a) = s0(a), a ∈ [0, τc], e(0, a) = e0(a), a > 0, Epo(0) = Epo0, GC(0) = GC0. (7)
The effort made on increasing the realism in the model is however achieved at the expense of loss in mathematical tractability. We point out that, without other restrictive assumptions, this model cannot be solved analytically. There-fore, the use of efficient methods that provide a numerical approach is the most suitable mathematical tool for studying the problem and, indeed, it is often the only one available. Besides, numerical methods have been successfully applied to structured models to replicate available field and/or laboratory data for a variety of different systems (e.g. [9, 11, 12, 13] and references therein). We completely describe, in section 4, the explicit second order numerical scheme, built “ad hoc”, which has been used to obtain the solutions of (4)–(7). And, finally, we perform a wide numerical experimentation which includes conver-gence, long-time integration with the simulation of the theoretical steady state and a comparison with real biological data in section 5.
3. Existence of a Positive Steady State
We focus in this section on the existence of steady states for system (4)– (6) when γ does not depend on time t, that is in normal erythropoiesis. From a biological point of view, system (4)–(6) is expected to possess in this case only one positive steady state, due to the constant positive flux HSC from the hematopoietic stem cell compartment.
Suppose (p∗(a), s∗(a), e∗(a), Epo∗, GC∗) is a steady state solution of system (4)–(6). Then, it satisfies dp∗ da(a) = − [α(Epo
∗) + σ(a, GC∗) + δ(a, Epo∗)] p∗(a), a ∈ (0, τ p), ds∗ da(a) = −α(Epo ∗)s∗(a), a ∈ (0, τ c), de∗ da(a) = −γ(a)e ∗(a), a > 0, (8) with p∗(0) = HSC + 2s∗(τ c), s∗(0) = Z τp 0 σ(a, GC∗)p∗(a)da, e∗(0) = A Z τp 0
δ(a, Epo∗)p∗(a)da,
(9)
and
kEpoEpo∗= f (E∗), kGCGC∗= g(E∗),
where
E∗= Z +∞
0
e∗(a)da. (10)
Let us introduce the following notations, ¯ α(E∗) := α(F (E∗)), ¯
σ(a, E∗) := σ(a, G(E∗)), a ∈ [0, τp],
¯
δ(a, E∗) := δ(a, F (E∗)), a ∈ [0, τp],
where F (E∗) := f (E∗)/kEpo and G(E∗) := g(E∗)/kGC. Consider the following
assumptions,
(H1) Suppose for all θ ∈ [0, τp],
∂Yβ(θ, Y ) > 0, ∀Y ≥ 0.
(H2) Suppose, for all θ ∈ [0, τp], there exists ¯Y > 0 such that
(
∂Yβ(θ, Y ) < 0, ∀Y ∈ [0, ¯Y ),
∂Yβ(θ, Y ) > 0, ∀Y ∈ ( ¯Y , +∞).
From a biological point of view, Assumption (H1) describes a situation in which as the number of erythrocytes increases progenitor apoptosis increases. This is expected, otherwise the erythrocyte population could be unbounded. As-sumption (H2) has a similar interpretation, but it also considers that for small erythrocytes counts, progenitor apoptosis rate becomes negligible compared to self-renewal and differentiation rates. This should theoretically allow a repopu-lation of the erythrocyte popurepopu-lation. Both assumptions are in agreement with biological knowledge.
The next proposition states the existence of a unique positive steady state. Proposition 1. Assume inequality
e−α(0)τc Z τp 0 σ(a, 0) exp − Z a 0 α(0) + σ(θ, 0) + δ(θ, 0)dθ da < 1 2 (11)
holds true. Then whether Assumption (H1) or (H2) is satisfied, system (4)–(6) has a unique steady state.
Proof. This proof is in two parts. We first show that system (8)–(10) has a solution if and only if condition (21) holds true. Then, as a second step, we show that under Assumption (H1) or (H2) condition (21) holds true as soon as (11) is satisfied.
Step 1. Solving system (8)–(9) gives
p∗(a) = exp − Z a 0 β(θ, E∗)dθ [HSC + 2s∗(τc)] , a ∈ [0, τp], (12)
Using now (15) in (14) one obtains e∗(a) = exp − Z a 0 γ(θ)dθ AΛ(E∗)Ψ(E∗, p∗), a > 0, with Λ(E∗) := Z τp 0 ¯ δ(θ, E∗) exp − Z θ 0 β(u, E∗)du ! dθ. (17) It follows that E∗:= Z +∞ 0
e∗(a)da = ξAΛ(E∗)Ψ(E∗, p∗), (18)
where ξ := Z +∞ 0 exp − Z a 0 γ(θ)dθ da > 0. From (15), one gets for a ∈ [0, τp]
¯
σ(a, E∗)p∗(a) = ¯σ(a, E∗) exp − Z a 0 β(θ, E∗)dθ Ψ(E∗, p∗). (19)
Introducing the following notations, µ(E∗) := e− ¯α(E∗)τc, ν(E∗) := Z τp 0 ¯ σ(a, E∗) exp − Z a 0 β(θ, E∗)dθ da, (20)
and using (16), we finally get from the integration of (19) [1 − 2µ(E∗)ν(E∗)]
Z τp
0
¯
σ(θ, E∗)p∗(θ)dθ = ν(E∗)HSC.
Hence, providing that 1 − 2µ(E∗)ν(E∗) 6= 0, we have from (16)
Ψ(E∗, p∗) = HSC + 2µ(E∗) ν(E
∗)HSC 1 − 2µ(E∗)ν(E∗) = HSC 1 − 2µ(E∗)ν(E∗), and from (18) Ψ(E∗, p∗) = E ∗ AξΛ(E∗).
This yields that system (4)–(6) has a steady state (p∗(a), s∗(a), e∗(a), Epo∗, GC∗), withR+∞
0 e
∗(a)da := E∗, if
1 − 2µ(E∗)ν(E∗) 6= 0 and E∗= A.HSC.ξ Λ(E
∗)
1 − 2µ(E∗)ν(E∗). (21)
Conversely, suppose there exists a constant X > 0 such that
1 − 2µ(X)ν(X) 6= 0 and X = A.HSC.ξ Λ(X)
Define K := HSC ν(X) 1 − 2µ(X)ν(X), and set z(a) = exp − Z a 0 β(θ, X)dθ [HSC + 2µ(X)K] , a ∈ [0, τp],
y(a) = exp (− ¯α(X)a) K, a ∈ [0, τc],
x(a) = exp − Z a 0 γ(θ)dθ A.Λ(X) [HSC + 2µ(X)K] , a > 0. Then, it is straightforward, using (20) and setting Epo∗ = F (X) and GC∗ = G(X), to check that (z, y, x) satisfies (8)–(9) and (10). In particular, using the definition of x and ξ one obtains
Z +∞
0
x(a)da = AξΛ(X) [HSC + 2µ(X)K] , and using the definitions of K and X one finally gets
Z +∞ 0 x(a)da = A.HSC.ξΛ(X) 1 + 2µ(X) ν(X) 1 − 2µ(X)ν(X) = X, so (10) is satisfied. This defines a steady state solution of system (4)–(6). Step 2. We now prove that, under assumptions (H1) or (H2), there exists a unique E∗> 0 satisfying (21). We first set
χ(Y ) := 1 − 2µ(Y )ν(Y ), Y ≥ 0, and focus on the problem
Find Y ≥ 0 such that χ(Y ) > 0 and cY =Λ(Y )
χ(Y ), (22)
where c := 1/(A.HSC.ξ) > 0. From (17) and (20), and the previous assump-tions on the funcassump-tions α, σ and δ (see section 2), we obtain
d dYΛ(Y ) = Z τp 0 " ∂Yδ(θ, Y ) −¯ Z θ 0 ∂Yβ(u, Y )du ! ¯ δ(θ, Y ) # e−R0θβ(u,Y )dudθ, and d dYν(Y ) = Z τp 0 " ∂Yσ(θ, Y ) −¯ Z θ 0 ∂Yβ(u, Y )du ! ¯ σ(θ, Y ) # e−R0θβ(u,Y )dudθ, and ¯δ(θ, Y ) > 0, ¯σ(θ, Y ) > 0, ∂Yδ(θ, Y ) < 0 and ∂¯ Yσ(θ, Y ) < 0.¯
First assume (H1) holds true. Then d
dYΛ(Y ) < 0 and d
A first consequence, since by definition dµ(Y )/dY < 0, is that d dYχ(Y ) = −2 ν(Y ) d dY µ(Y ) + µ(Y ) d dYν(Y ) > 0.
Hence, the function χ is increasing on [0, +∞). Moreover, (11) implies that limY →+∞χ(Y ) > 0. Hence, there exists Y0 ≥ 0 such that χ(Y ) > 0 for Y ∈
(Y0, +∞) and χ(Y0) ≥ 0.
Moreover, from (H1) and as long as χ(Y ) > 0,
d dY Λ(Y ) χ(Y ) = χ(Y ) d dYΛ(Y ) − Λ(Y ) d dY χ(Y ) χ(Y )2 < 0.
Hence, the function Λ(Y )/χ(Y ) is decreasing on the domain (Y0, +∞), and
Λ(Y0)/χ(Y0) > 0. Consequently (22) has a unique solution Y > Y0.
Assume (H2) holds true. Similarly to the above reasoning, Λ(Y ) and ν(Y ) are decreasing on ( ¯Y , +∞), and the function Λ(Y )/χ(Y ) is decreasing for Y > max{Y0, ¯Y }. We then conclude similarly to the previous case to the existence
and uniqueness of Y > max{Y0, ¯Y } satisfying (22).
Under assumptions (H1) or (H2) we can then define a unique steady state to our problem. This concludes the proof.
Let us briefly comment on (11). This condition implies that no more than half of the progenitor population can self-renew in order to obtain a stationary solution. Indeed, the term exp −R0aα(0) + σ(θ, 0) + δ(θ, 0)dθ describes the survival rate of differentiating progenitors, σ(a, 0) refers to the maximal fraction of the population that will self-renew, and exp(−α(0)τc) is the maximal survival
fraction of self-renewing progenitors. Inequality (11) is satisfied with our choices for α, σ and δ (see Equations (1), (2), (3)). If inequality (11) is not satisfied, then one may hypothesize the non-existence of a stationary solution and the unboundedness of the cell population.
4. Numerical Scheme
We introduce a new numerical method to simulate the system (4)-(7). It is based on the integration along the characteristic curves with the use of a representation formula of the solution [1].
The existence of a maximum age of differentiation, τp, makes the
differen-tiation rate δ unbounded. This fact has been circumvented in [7, 14] with the use of an exact integration value at the end of the age-differentiation interval. However, this method is only possible when such an exact value is known, and this only happens with specific functions. In general, it is necessary to introduce a new different dependent function as in [30]. We assume, from now on, that age and Epo dependences of the function δ are independent, and we write, as in (3), δ(a, Epo) = δ1(a) + δ2(Epo).
and, accordingly, p0(a) = Π(a) q0(a). The first equation of (4) is replaced with
∂tq + ∂aq = −(α(Epo(t)) + δ2(Epo(t)) + σ(a, GC(t))) q, a ∈ (0, τp). (23)
This change also modifies the boundary condition (5), that writes q(t, 0) = HSC + 2s(t, τc), s(t, 0) = Z τp 0
σ(a, GC(t)) Π(a) q(t, a)da, e(t, 0) = A
Z τp
0
δ(a, Epo(t)) Π(a) q(t, a)da.
(24)
The unbounded age domain for erythrocytes is another circumstance that must be avoided in the numerical integration. Thus, we introduce a maximum age Amaxfor erythrocytes as in [7], that would be large enough to simulate the
dynamics of the model. This means that the numerical method approaches the evolution of a “truncated” version of problem (23)-(24) and (6)-(7), which in turn is an approximation to the solution of the original problem (see [10]).
The formula we use in the numerical method is based on the theoretical integration along the characteristics, which gives us the next representation of the solution to the new system,
q(t∗+ h, a∗+ h) = q(t∗, a∗) exp − Z h 0 β∗(a∗+ x, Epo(t∗+ x), GC(t∗+ x)) dx ! , a∗+ h ≤ τp, s(t∗+ h, a∗+ h) = s(t∗, a∗) exp − Z h 0 α(Epo(t∗+ x)) dx ! , a∗+ h ≤ τc, e(t∗+ h, a∗+ h) = e(t∗, a∗) exp − Z h 0 γ(t∗+ x) dx ! , a∗+ h ≤ Amax, (25)
where β∗(a, E, G) = α(E) + δ2(E) + σ(a, G). Note that, in this new layout, we
have two different problems: solving (25), which provides the solution to the population densities, and integrating (6), which gives the solution to the growth factor concentrations. We use discretization procedures in order to solve them. We consider the numerical integration of model (23)-(24) and (6)-(7) along the time interval [0, T ]. Thus, given a positive integer Jc, we define h = τc/Jc,
Jp = [τp/h], Jmax = Amax/h and N = [T /h] (it is usual that τc and τp would
be proportional, and Amax might be chosen to be proportional to τc, these
restrictions do not play an important role in the numerical integration). We introduce the discrete time levels tn = n h, 0 ≤ n ≤ N , and the grid nodes
aj = j h, 0 ≤ j ≤ Jmax. Then, we denote the numerical approximations to the
theoretical solutions by: Qn
j to qnj = q(aj, tn), 0 ≤ j ≤ Jp; Sjn to snj = s(aj, tn),
0 ≤ j ≤ Jc; Ijn to enj = e(aj, tn), 0 ≤ j ≤ Jmax; On to on = Epo(tn); and Gn
to gn= GC(tn); 0 ≤ n ≤ N . We propose a new method to obtain them: (i) we
the grid restriction of functions q0, s0, e0and the values Epo0and GC0, and (ii)
the numerical solution at a new time level is described in terms of the previous one by means of a two steps scheme.
Such a general time level is obtained by means of the following second-order discretization of (24)-(25) and (6). For, 0 ≤ n ≤ N − 1,
Qn+1j+1 = Qnj exp (−h (β∗(aj, On, Gn) + β∗(aj+1, On+1,∗, Gn+1,∗))/2),
0 ≤ j ≤ Jp− 1,
Sj+1n+1= Sjn exp (−h (α(On) + α(On+1,∗))/2), 0 ≤ j ≤ Jc− 1,
Ij+1n+1= In
j exp (−h (γ(aj, tn) + γ(aj+1, tn+1))/2), 0 ≤ j ≤ Jmax− 1,
with Qn+10 = HSC + 2 SJn+1 c , Sn+10 = Qh(σn+1· Π · Qn+1), I0n+1 = A Qh(δn+1· Π · Qn+1), and On+1 = On+ h (f (Q h(In)) + f (Qh(In+1,∗)) − kEpo(On+ On+1,∗))/2, Gn+1 = Gn+ h (g(Q h(In)) + g(Qh(In+1,∗)) − kGC(Gn+ Gn+1,∗))/2.
Where the intermediate step is given by
Qn+1,∗j+1 = Qnj exp (−h β∗(aj, On, Gn)), 0 ≤ j ≤ Jp− 1,
Sj+1n+1,∗ = Sjn exp (−h α(On)), 0 ≤ j ≤ Jc− 1,
Ij+1n+1,∗ = Ijn exp (−h γ(aj, tn)), 0 ≤ j ≤ Jmax− 1,
with Qn+1,∗0 = HSC + 2 SJn+1,∗ c , I0n+1,∗ = A Qh(δn+1,∗· Π · Qn+1,∗), and On+1,∗ = On+ h (f (Qh(In)) − kEpoOn), Gn+1,∗ = Gn+ h (g(Qh(In)) − kGCGn).
One can note that initializing S0n+1,∗ is not necessary. In these formulae, Qh(Vn) represents a quadrature rule to approximate the integral on the
inter-val [0, Amax] (approximation to the total number of erythrocytes E(t)) or [0, τp]
(discretization of the boundary condition (24)) of the function with grid values Vn, with different number of nodes each case, this fact is indicated by the vector
so it does not cause confusion. In this case Πj = Π(aj), δn+1j = δ(aj, On+1),
δn+1,∗j = δ(aj, On+1,∗), σn+1j = σ(aj, Gn+1), 0 ≤ j ≤ Jp, and δn+1· Π · Qn+1,
δn+1,∗· Π · Qn+1,∗, σn+1· Π · Qn+1, denote the componentwise product of
5. Numerical Experiments
We present in this section the analysis performed to experimentally check the behavior of the proposed numerical method with different test problems. The first test consists in confronting predictions of the theoretical analysis (steady state values) with the method’s ability to generate a good approximation to the solution of the problem, even in the case of a long time integration allowing to approach steady state values. The second test aims at showing the capacity to obtain an accurate approximation when confronting the model to real biological data: we simulate an experimentally-induced anemia in mice and the response of the organism. The accuracy in the numerical integration is more important in this case than the long time integration.
In order to perform each test, we have used the functions α, σ and δ defined in (1)-(3).
The average lifespan of an erythrocyte is known to be 40 days in mice [21], so we assume a constant mortality rate, γ(a, t) = γ = 1/40 d−1.
Finally, the following Hill functions have been used in (6),
f (E) = Cf θnf f θnf f + Enf , g(E) = Cg θng g θng g + Eng , (26)
where (avoiding the subscripts) C > 0 denotes the maximum of the function, θ > 0 a threshold value, and n > 1 a sensitivity parameter.
Parameter values are listed in Tables 1 and 3. Some values have been ob-tained from existing values in the literature, others have been estimated using a classical minimization of a weighted residual sum of squares in order to fit biological measurements (this is indicated by [NS] in Tables 1 and 3).
Test Problem 1. The following numerical experiment shows the efficacy of the numerical method in simulating our model. To this end, we compare the be-havior of the numerical solution with the solution provided by the theoretical analysis in section 3. In this section, existence of a unique positive stationary population density was shown, and we shall use it in the comparison of the numerical order of convergence of our scheme. We will show the optimal rate of convergence obtained with the numerical method.
We take the functions in (1)-(3) and (26) and use the parameter values in Table 1.
Table 1: Parameter values. Sources of parameter values are given between brackets, with [NS] = values obtained by performing numerical simulations. The values of θα, θσ, Cf and
Cg are respectively given by θ∗α= log10(2Epo∗), θσ∗ = log10(3GC∗), Cf∗= kepoEpo ∗(θnf f + (E∗)nf)/θnf f , C ∗ g = kgcGC∗(θgng+ (E∗)ng)/θ ng g . (N.U. = no unit)
Par. Value Unit Par. Value Unit
Cell Populations [21], [NS] Growth factors [20]
A 28 N.U. k
Epo 5.55 day−1
HSC 104 cells.g−1.d−1 Epo∗ 5.7 mU.µL−1
γ 1/40 d−1 kGC 11.1 d−1
E∗ 1.51 × 107 cells GC∗ 44.6 mU.µL−1
Cell Cycle and Differentiation Durations [21]
τp 4 d τc 1 d
Apoptosis Rate [21], [NS] Self-Renewal Rate [21], [NS]
Cα 0.6 d−1 Cσ 1 d−2 θα θ∗α mU.µL−1 θσ θσ∗ mU.µL−1 nα 2 N.U. nσ 2 N.U. Differentiation Rate [21], [NS] Cδ1 1.15 d −1 θ δ 1 mU.µL−1 Cδ2 0.1 d −1 n δ 2 N.U.
Epo Production Rate [20], [NS] GC Production Rate [20], [NS]
Cf Cf∗ mU.µL
−1.d−1 C
g Cg∗ mU.µL−1.d−1
θf 0.7E∗ cells.g−1 θg 0.4E∗ cells.g−1
nf 7 N.U. ng 6 N.U.
which satisfy the first compatibility condition. Note that (27) is well defined. We complete the initial condition with Epo0 = 5 mU.µL−1 and GC0 = 44
mU.µL−1.
We do not know the analytical solution to the problem then, in order to compare with the numerical solution, we use the value of the steady state of the problem, given by the following formulae
s∗(a) = HSC σ(0, GC ∗) i 1 1 − 2 σ(0, GC∗) i 1e−α(a,Epo ∗) τ ce −α(a,Epo∗) a , a ≤ τc, (28) e∗(a) = A HSC Cδ1i−1+ δ2(0, Epo ∗) i 0 1 − 2 σ(0, GC∗) i 1e−α(a,Epo∗) τc e−γ a, a ≤ τp, (29) p∗(a) = HSCe −a (α(a,Epo∗)+δ 2(a,Epo∗)+(1−a/(2 τp)) σ(0,GC∗)) 1 − 2 σ(0, GC∗) i 1e−α(a,Epo ∗) τ c 1 − a τp τpCδ1 , a ≤ τp, (30) q∗(a) = HSCe −a (α(a,Epo∗)+δ 2(a,Epo∗)+(1−a/(2 τp)) σ(0,GC∗)) 1 − 2 σ(0, GC∗) i 1e−α(a,Epo ∗) τ c , a ≤ τp, (31) where, for r = −1, 0, 1, ir= Z τp 0 1 − a τp Cδ1τp+r
h Ehp Es h E e h E Epo h E GC h
3.125e-2 5.6446e-3 3.4988e-3 5.1756e-1 2.3044e-6 1.6660e-5 1.5625e-2 1.4187e-3 8.8050e-4 1.3073e-1 5.7095e-7 4.1277e-6
1.99 1.99 1.99 2.01 2.01
7.8125e-3 3.5520e-4 2.2052e-4 3.2773e-2 1.4241e-7 1.0296e-6
2.00 2.00 2.00 2.00 2.00
3.9063e-3 8.8846e-5 5.5164e-5 8.2001e-3 3.5589e-8 2.5729e-7
2.00 2.00 2.00 2.00 2.00
1.9531e-3 2.2236e-5 1.3806e-5 2.0524e-3 8.9049e-9 6.4378e-8
2.00 2.00 2.00 2.00 2.00
Table 2: Test problem 1. Error and numerical convergence order, for T = 1600.
We have carried out an extensive numerical experimentation with differ-ent final-times T , step-sizes h and maximum erythrocites age Amax values. We
observe that Amax= 1600 is enough to obtain a good approximation to the
non-truncated problem and T = 1600 produces a sufficiently long time integration in order to provide a numerical approximation to the steady state.
In Table 2, we present the results obtained with the method for different values of the step size. For each h, we compare at the final time T = 1600, the computed numerical solution Unh, with the steady state un
h, where U n
h and unh
represent, respectively, Pnh, Snh, Inh, On and Gn and pn h, s n h, e n h, Epo ∗and GC∗,
0 ≤ n ≤ N . In each row of Table 2, the upper number shows the maximum error with different step sizes h (first column); that is
Eu h = max 0≤n≤NkU n h− u nk ∞,
and the lower number the numerical order of convergence, which we compute with the formula
su2 h= log(E u 2 h/E u h) log(2) .
Results in Table 2 clearly confirm the expected second-order of convergence. Test Problem 2. The second test deals with the reproduction of biological mea-surements. Data come from an experiment of induced anemia in mice, intro-duced in [21]. We assume that the initial condition of the system is given by the steady state (28)-(31), then we simulate the anemia (induced by a drug called phenylhydrazine that kills red blood cells upon injection on days 0 and 1, see Figure 2) and we observe the reaction of the organism that leads to full recovery within few days (see Figure 2). Thus, we are not interested on a long time integration but rather on an accurate description of transient dynamics.
0 5 10 15 20 25 30 35 40 45 10 15 20 25 30 35 40 45 50 55 60 Time (days) Hematocrit (%)
Figure 2: Experimental and Simulated Hematocrits. Experimental data are given by the blue dashed line, with error bars on every experimental time point. The simulated hematocrit is given by the red curve. Mice are rendered anemic by two consecutive injections of phenylhydrazine on days 0 and 1. The hematocrit shows a strong fall following the anemia (the hematocrit reaches very low values), then it rapidly increases to reach a high value, and then returns to the equilibrium. The simulated hematocrit is able to correctly capture all these features. All parameter values are given by Tables 1 and 3.
hematocrit HCT is computed using the formula in [21],
HCT (t) = E(t)
E(t) + E∗(1 − HCT∗)/HCT∗,
where E(t) is the total number of erythrocytes and HCT∗ is the steady state value of the hematocrit, obtained through experimental data (in mice, HCT∗ is around 45% − 50%, see Figure 2).
Again, we use the functions in (1)-(3) and (26) to perform the simulations. We have carried out an extensive numerical experimentation with different final-times T , step-sizes h and values of the parameters involved in the problem. We observe that T = 200 and h = 7.8125e−3 provide sufficiently accurate approximations to the theoretical stationary state.
Table 3: Parameter values. Sources of parameter values are given between brackets, with [NS] = values obtained by performing numerical simulations. (N.U. = no unit)
Par. Value Unit Par. Value Unit
Steady States [21], [NS] E∗ 1.72 × 107 cells HCT∗ 48% N.U. Phenylhydrazine Injections [NS] R 2 d−1 K 9 d−1 tf 1 d m 4 N.U. t2 1 d
mortality rate γ(a, t), as follows
γ(t, a) = γ (1 + phz(a, t, 0, tf) + phz(a, t, t2, tf)) ,
where the function phz accounts for the effect of phenylhydrazine, as follows
phz(a, t, ti, tf)= 0, t ≤ ti or a ≤ t − ti, K(ti+ tf− t) m tm f + R, ti≤ t ≤ ti+ tf and a ≥ t − ti, R, ti+ tf ≤ t and a ≥ t − ti,
with ti the time of the injection, tf the duration of the phenylhydrazine effect
in the blood, R a nonnegative residual effect, K a positive constant which determines the maximum effect of the phenylhydrazine injection, and m an exponent associated with the clearance of the phenylhydrazine while in blood.
In order to compare the model simulations with the experimental data, we then reproduced the experimental protocol consisting in two injections of phenylhydrazine, with the initial injection occurring at time t = 0 and the second injection at time t2 = 1 day. Since experimentally both injections are
similar (same dose, same route of injection), we assumed that they both have the same effects and we used the same parameter values for the function phz, except for the value of ti. Additional parameter values are shown in Table 3.
Figure 2 shows the simulation performed numerically with γ = 1/√40, pa-rameter data in Tables 1, and 3, and associated experimental data. One can appreciate how the numerical solution reproduces the behavior of the phenyl-hydrazine action in inducing anemia and the recovery of the organism.
6. Conclusions
In order to perform a more complex analysis of the model, we needed to introduce numerical approximation to the solution. Therefore, we presented an “ad hoc” new numerical method to generate a simulated solution of the system. We have studied numerically both its accuracy and its potential to simulate real biological behaviors, based on experimental data. On one hand, we have shown numerically a second order of convergence. On the other hand, the method proved able to accurately replicate an experimental protocol, consisting in injecting mice with a substance that kills red blood cells and induces anemia, which has been modeled following [8]. Thus, the numerical scheme shows a second order convergence in a finite time integration, a good behavior in a long-time one and it is able to simulate and replicate biological data.
Aknowledgements
FC work has been supported by ANR grant ProCell ANR-09-JCJC-0100-01. OA and JCLM have been supported in part by project MTM2014-56022-C2-2-P of the Spanish Ministerio de Econom´ıa y Competitividad and European FEDER Funds, and by Consejer´ıa de Educaci´on, JCyL, project VA191U13.
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